Theorem cnmpt1vsca 17876

Description: Continuity of scalar multiplication; analogue of cnmpt12f 17360 which cannot be used directly because .s is not a function. (Contributed by Mario Carneiro, 5-Oct-2015.)

Hypotheses

Ref	Expression
tlmtrg.f	|- F = (Scalar ` W )
cnmpt1vsca.t	|- .x. = ( .s ` W )
cnmpt1vsca.j	|- J = ( TopOpen ` W )
cnmpt1vsca.k	|- K = ( TopOpen ` F )
cnmpt1vsca.w	|- ( ph -> W e. TopMod )
cnmpt1vsca.l	|- ( ph -> L e. (TopOn ` X ) )
cnmpt1vsca.a	|- ( ph -> ( x e. X |-> A ) e. ( L Cn K ) )
cnmpt1vsca.b	|- ( ph -> ( x e. X |-> B ) e. ( L Cn J ) )

Assertion

Ref	Expression
cnmpt1vsca	|- ( ph -> ( x e. X |-> ( A .x. B ) ) e. ( L Cn J ) )

Distinct variable groups:   x, F   x, J   x, K   x, L   ph, x   x, W   x, X

Allowed substitution hints:   A( x)   B( x)   .x. ( x)

Proof of Theorem cnmpt1vsca

Step	Hyp	Ref	Expression
1		cnmpt1vsca.l	. . . . . . 7 |- ( ph -> L e. (TopOn ` X ) )
2		cnmpt1vsca.w	. . . . . . . . 9 |- ( ph -> W e. TopMod )
3		tlmtrg.f	. . . . . . . . . 10 |- F = (Scalar ` W )
4	3	tlmscatps 17873	. . . . . . . . 9 |- ( W e. TopMod -> F e. TopSp )
5	2, 4	syl 15	. . . . . . . 8 |- ( ph -> F e. TopSp )
6		eqid 2283	. . . . . . . . 9 |- ( Base ` F ) = ( Base ` F )
7		cnmpt1vsca.k	. . . . . . . . 9 |- K = ( TopOpen ` F )
8	6, 7	istps 16674	. . . . . . . 8 |- ( F e. TopSp <-> K e. (TopOn ` ( Base ` F ) ) )
9	5, 8	sylib 188	. . . . . . 7 |- ( ph -> K e. (TopOn ` ( Base ` F ) ) )
10		cnmpt1vsca.a	. . . . . . 7 |- ( ph -> ( x e. X |-> A ) e. ( L Cn K ) )
11		cnf2 16979	. . . . . . 7 |- ( ( L e. (TopOn ` X ) /\ K e. (TopOn ` ( Base ` F ) ) /\ ( x e. X |-> A ) e. ( L Cn K ) ) -> ( x e. X |-> A ) : X --> ( Base ` F ) )
12	1, 9, 10, 11	syl3anc 1182	. . . . . 6 |- ( ph -> ( x e. X |-> A ) : X --> ( Base ` F ) )
13		eqid 2283	. . . . . . 7 |- ( x e. X |-> A ) = ( x e. X |-> A )
14	13	fmpt 5681	. . . . . 6 |- ( A. x e. X A e. ( Base ` F ) <-> ( x e. X |-> A ) : X --> ( Base ` F ) )
15	12, 14	sylibr 203	. . . . 5 |- ( ph -> A. x e. X A e. ( Base ` F ) )
16	15	r19.21bi 2641	. . . 4 |- ( ( ph /\ x e. X ) -> A e. ( Base ` F ) )
17		tlmtps 17870	. . . . . . . . 9 |- ( W e. TopMod -> W e. TopSp )
18	2, 17	syl 15	. . . . . . . 8 |- ( ph -> W e. TopSp )
19		eqid 2283	. . . . . . . . 9 |- ( Base ` W ) = ( Base ` W )
20		cnmpt1vsca.j	. . . . . . . . 9 |- J = ( TopOpen ` W )
21	19, 20	istps 16674	. . . . . . . 8 |- ( W e. TopSp <-> J e. (TopOn ` ( Base ` W ) ) )
22	18, 21	sylib 188	. . . . . . 7 |- ( ph -> J e. (TopOn ` ( Base ` W ) ) )
23		cnmpt1vsca.b	. . . . . . 7 |- ( ph -> ( x e. X |-> B ) e. ( L Cn J ) )
24		cnf2 16979	. . . . . . 7 |- ( ( L e. (TopOn ` X ) /\ J e. (TopOn ` ( Base ` W ) ) /\ ( x e. X |-> B ) e. ( L Cn J ) ) -> ( x e. X |-> B ) : X --> ( Base ` W ) )
25	1, 22, 23, 24	syl3anc 1182	. . . . . 6 |- ( ph -> ( x e. X |-> B ) : X --> ( Base ` W ) )
26		eqid 2283	. . . . . . 7 |- ( x e. X |-> B ) = ( x e. X |-> B )
27	26	fmpt 5681	. . . . . 6 |- ( A. x e. X B e. ( Base ` W ) <-> ( x e. X |-> B ) : X --> ( Base ` W ) )
28	25, 27	sylibr 203	. . . . 5 |- ( ph -> A. x e. X B e. ( Base ` W ) )
29	28	r19.21bi 2641	. . . 4 |- ( ( ph /\ x e. X ) -> B e. ( Base ` W ) )
30		eqid 2283	. . . . 5 |- ( .s f ` W ) = ( .s f ` W )
31		cnmpt1vsca.t	. . . . 5 |- .x. = ( .s ` W )
32	19, 3, 6, 30, 31	scafval 15646	. . . 4 |- ( ( A e. ( Base ` F ) /\ B e. ( Base ` W ) ) -> ( A ( .s f ` W ) B ) = ( A .x. B ) )
33	16, 29, 32	syl2anc 642	. . 3 |- ( ( ph /\ x e. X ) -> ( A ( .s f ` W ) B ) = ( A .x. B ) )
34	33	mpteq2dva 4106	. 2 |- ( ph -> ( x e. X |-> ( A ( .s f ` W ) B ) ) = ( x e. X |-> ( A .x. B ) ) )
35	30, 20, 3, 7	vscacn 17868	. . . 4 |- ( W e. TopMod -> ( .s f ` W ) e. ( ( K tX J ) Cn J ) )
36	2, 35	syl 15	. . 3 |- ( ph -> ( .s f ` W ) e. ( ( K tX J ) Cn J ) )
37	1, 10, 23, 36	cnmpt12f 17360	. 2 |- ( ph -> ( x e. X |-> ( A ( .s f ` W ) B ) ) e. ( L Cn J ) )
38	34, 37	eqeltrrd 2358	1 |- ( ph -> ( x e. X |-> ( A .x. B ) ) e. ( L Cn J ) )

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1623   e. wcel 1684   A.wral 2543   e. cmpt 4077   -->wf 5251   ` cfv 5255  (class class class)co 5858   Basecbs 13148  Scalarcsca 13211   .scvsca 13212   TopOpenctopn 13326   .s fcscaf 15628  TopOnctopon 16632   TopSpctps 16634   Cn ccn 16954   tX ctx 17255  TopModctlm 17840

This theorem is referenced by:  tlmtgp  17878

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pow 4188  ax-pr 4214  ax-un 4512

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-sbc 2992  df-csb 3082  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-pw 3627  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-iun 3907  df-br 4024  df-opab 4078  df-mpt 4079  df-id 4309  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-fv 5263  df-ov 5861  df-oprab 5862  df-mpt2 5863  df-1st 6122  df-2nd 6123  df-map 6774  df-slot 13152  df-base 13153  df-topgen 13344  df-scaf 15630  df-top 16636  df-bases 16638  df-topon 16639  df-topsp 16640  df-cn 16957  df-tx 17257  df-tmd 17755  df-tgp 17756  df-trg 17842  df-tlm 17844